| import OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingParameters |
| import OAI.Combinatorics.ContingencyTables.Machines.ListWordProgram |
|
|
| namespace OAI |
|
|
| /-! # The exact dyadic gate that selects the rare correction branch -/ |
|
|
| namespace ContingencyTables.ProfilePrograms |
| open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream DensePrograms Polynomial |
|
|
| /-- All zero bits select correction; every other word selects the ordinary draw. -/ |
| def rareWord (z : CompletionMargins×List Bool) : Bool := decide (StackAddition.value z.2=0) |
|
|
| def rareWordRealizer : Realizer rareWord := |
| composition (f := fun z : CompletionMargins×List Bool => (StackAddition.value z.2,0)) |
| (g := fun p : ℕ×ℕ => decide (p.1=p.2)) |
| (pair (composition second bitValue) (constant 0)) natEqual |
|
|
| lemma polynomial_rareWord : PolynomialTime rareWordRealizer := |
| polynomial_composition (polynomial_pair (polynomial_composition polynomial_second polynomial_bitValue) |
| (polynomial_constant 0)) polynomial_natEqual |
|
|
| def rareBranch (z : CompletionMargins) : Experiment Bool := |
| FiniteWordExecution.experiment exactPrecision rareWord z |
|
|
| noncomputable def rareBranchRealizer : OnlineTree.Realizer rareBranch := |
| FiniteWordExecution.realizer exactPrecisionRealizer rareWordRealizer |
|
|
| lemma work_rareBranch : OnlineTree.WorkBound rareBranchRealizer exactParameterMeasure := by |
| apply OnlineTree.work_remeasure |
| (FiniteWordExecution.work_realizer polynomial_exactPrecision polynomial_rareWord) X |
| intro z |
| simp only [FiniteWordExecution.wordMeasure,exactParameterMeasure,eval_X] |
| omega |
|
|
| lemma rareBranch_probability (z : CompletionMargins) : |
| (rareBranch z).mean (fun rare => if rare then 1 else 0)=1/(2:ℚ)^(exactPrecision z) := by |
| rw [rareBranch,FiniteWordExecution.experiment_mean] |
| simp only [rareWord,decide_eq_true_eq,bitValue_eq_word] |
| let zeroWord : Fin (exactPrecision z) → Bool := |
| (FairBits.binaryWordEquiv (exactPrecision z)).symm ⟨0,by positivity⟩ |
| have he (bits : Fin (exactPrecision z) → Bool) : |
| (FairBits.binaryWordEquiv (exactPrecision z) bits).val=0 ↔ bits=zeroWord := by |
| constructor |
| · intro h |
| apply (FairBits.binaryWordEquiv (exactPrecision z)).injective |
| apply Fin.ext |
| simpa only [zeroWord,Equiv.apply_symm_apply] using h |
| · intro h |
| rw [h] |
| simp only [zeroWord,Equiv.apply_symm_apply] |
| simp only [he,average,Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
| |
| end ContingencyTables.ProfilePrograms |
| |
| end OAI |
| |
